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Chapter 2: Problem 22

In a medical study patients are classified in 8 ways according to whether they have blood type \(A B^{+}\), \(A B \sim, A^{+}, A^{-}, B^{+}, B \sim, O^{+},\) or \(O^{-},\) and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classified.

Short Answer

Expert verified
The number of ways a patient can be classified is 24.

Step by step solution

01

Determine the number of blood types

First, count the number of distinct blood types. Here we have 8 types: \(A B^{+}\), \(A B \sim\), \(A^{+}\), \(A^{-}\), \(B^{+}\), \(B \sim\), \(O^{+}\), and \(O^{-}\).
02

Determine the number of blood pressure categories

Next, count the number of blood pressure categories. We have low, normal, and high blood pressure - giving us 3 categories.
03

Apply the multiplication principle

Multiply the number of blood types by the number of blood pressure categories to find the total number of classifications. This is done based on the multiplication principle which states that if there are m ways to do one thing, and n ways to do another, then there are m*n ways to do both.
04

Calculate the total number of classifications

Based on the multiplication from the previous step (8 blood types * 3 blood pressure categories), the total number of classifications for a patient is 24.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Blood Type Classification
Blood type classification is a system used to categorize the blood of different people based on the presence or absence of certain antigens. These antigens are substances that can trigger an immune response if they are foreign to the body. Blood types are inherited and remain the same throughout a person's life. In general, there are four major blood group systems:
  • Type A: which has antigen A on the surface of red blood cells.
  • Type B: which has antigen B.
  • Type AB: which has both antigens A and B.
  • Type O: which has neither antigen.
Each of these groups can be further specified by the presence (+) or absence (-) of the Rh factor, making the blood types classified as A+, A-, B+, B-, AB+, AB-, O+, and O-. Understanding these differences is crucial for blood transfusions and medical treatments to prevent adverse reactions.
Blood Pressure Categories
Blood pressure is an essential physiological parameter. It is the force exerted by circulating blood on the walls of blood vessels. It is a crucial indicator of cardiovascular health. Blood pressure is usually expressed in terms of the systolic (maximum) over diastolic (minimum) pressures, for example, 120/80 mmHg. For classification purposes, blood pressure can be divided into three main categories:
  • Low Blood Pressure: Also known as hypotension, generally having readings lower than 90/60 mmHg.
  • Normal Blood Pressure: Ranges from 90/60 mmHg to about 120/80 mmHg.
  • High Blood Pressure: Also referred to as hypertension, generally having readings exceeding 140/90 mmHg.
Different categories help clinicians plan appropriate treatments, monitor changes, and manage patient risks effectively.
Multiplication Principle
The multiplication principle is a fundamental concept in probability and combinatorics. It is used to determine the number of possible outcomes in a series of events. If one event can occur in "m" ways and a second independent event can occur in "n" ways, then the number of ways both events can occur is the product m*n.
This principle is simple yet powerful. It is particularly useful in calculating outcomes in complex situations. It is applied in many areas such as counting problems, probability calculations, or logistic operations. For example, in the given exercise, the multiplication principle helps to determine the number of ways patients can be classified by combining the number of blood types with the number of blood pressure categories, resulting in 8 x 3 = 24 possible classifications.

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Most popular questions from this chapter

Exercise and diet are being studied as possible substitutes for medication to lower blood pressure. Three groups of subjects will be used to study the effect of exercise. Group one is sedentary while group two walks and group three swims for 1 hour a day. Half of each of the three exercise groups will be on a salt-free diet. An additional group of subjects will not exercise nor restrict their salt, but will take the standard medication. Use \(Z\) for sedentary, \(W\) for walker, \(S\) for swimmer, \(Y\) for salt, \(N\) for no salt, \(M\) for medication, and \(F\) for medication free. (a) Show all of the elements of the sample space \(S\). (b) Given that \(A\) is the set of non-medicated subjects and \(B\) is the set of walkers, list the elements of \(A \cup B\) (c) List the elements of \(A\) n \(B\).

In a regional spelling bee, the 8 finalists consist of 3 boys and 5 girls. Find the number of sample points in the sample space \(S\) for the number of possible orders at the conclusion of the contest for (a) all 8 finalists; (b) the first 3 positions.

One overnight case contains 2 bottles of aspirin and 3 bottles of thyroid tablets. A second tote bag contains 3 bottles of aspirin, 2 bottles of thyroid tablets, and I bottle of laxative tablets. If 1 bottle of tablets is taken at random from each piece of luggage, find the probability that (a) both bottles contain thyroid tablets: (b) neither bottle contains thyroid tablets; (c) the 2 bottles contain different tablets.

A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. From past experience it is known that \(20 \%:\) of the clients are assigned rooms at the Ramada Inn, \(50 \%\) at. the Sheraton, and \(30 \%\) at the Lake view Motor Lodge. If the plumbing is faulty in \(5 \%\) of the rooms at the Ramada Inn, in \(4 \%\) of the rooms at the Sheraton, and in \(8 \%\) of the rooms at the Lakeview Motor Lodge, what is the probability that (a) a client will be assigned a room with faulty plumbing? (b) a person with a room having faulty plumbing was assigned accommodations at the Lakeview Motor Lodge?

The probability that an automobile being filled with gasoline will also need an oil change is 0.25 ; the probability that it needs a new oil filter is 0.40 ; and the probability that both the oil and filter need changing is 0.14 (a) If the oil had to be changed, what is the probability that a new oil filter is needed? (b) If a new oil filter is needed, what is the probability that the oil has to be changed?
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